R E S E A R C H
Open Access
Fixed points of monotone nonexpansive
mappings with a graph
Monther Rashed Alfuraidan
**Correspondence:
monther@kfupm.edu.sa Department of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran, 31261, Saudi Arabia
Abstract
In this paper, we study the existence of fixed points of monotone nonexpansive mappings defined in Banach spaces endowed with a graph. This work is a continuity of the previous results of Ran and Reurings, Nietoet al., and Jachimsky done for contraction mappings defined in metric spaces endowed with a graph.
MSC: Primary 06F30; 46B20; 47E10
Keywords: Banach space; directed graph; fixed point; monotone nonexpansive mappings
1 Introduction
Banach’s contraction principle [] is remarkable in its simplicity, yet it is perhaps the most widely applied fixed point theorem in all of analysis. This is because the contractive condi-tion on the mapping is simple and easy to test, because it requires only a complete metric space for its setting, and because it is a powerful result with a wide range of applications, including iterative methods for solving linear, nonlinear, differential, integral, and differ-ence equations. Due to its applications in mathematics and other related disciplines, the Banach contraction principle has been generalized in many directions. Recently a version of this theorem was given in partially ordered metric spaces [, ] and in metric spaces endowed with a graph [–].
In this paper, we study the case of nonexpansive mappings defined in Banach spaces endowed with a graph. Nonexpansive mappings are those which have Lipschitz constant equal to . The fixed point theory for such mappings is rich and varied. It finds many appli-cations in nonlinear functional analysis []. It is worth mentioning that such investigation is, to the best of our knowledge, new and was never carried out. This work was inspired by [].
2 Graph basic definitions
The terminology of graph theory instead of partial ordering gives a wider picture and yields interesting generalization of the Banach contraction principle. In this section, we give the basic graph theory definitions and notations which will be used throughout.
LetGbe a directed graph (digraph) with a set of verticesV(G) and a set of edgesE(G) containing all the loops,i.e., (x,x)∈E(G) for anyx∈V(G). We also assume thatGhas no parallel edges (arcs), and so we can identifyGwith the pair (V(G),E(G)). Our graph theory notations and terminology are standard and can be found in all graph theory books
such as [] and []. Moreover, we may treatGas a weighted graph (see [], p.]) by assigning to each edge the distance between its vertices. ByG–we denote the conversion
of a graphG,i.e., the graph obtained fromGby reversing the direction of edges. Thus we have
EG–=(y,x)|(x,y)∈E(G).
A digraphGis called an oriented graph if whenever (u,v)∈E(G), then (v,u) /∈E(G). The letterGdenotes the undirected graph obtained fromGby ignoring the direction of edges. Actually, it will be more convenient for us to treatGas a directed graph for which the set of its edges is symmetric. Under this convention,
E(G) =E(G)∪EG–.
We call (V,E) a subgraph of G ifV⊆V(G), E⊆E(G) and for any edge (x,y)∈E, x,y∈V.
Ifxandyare vertices in a graphG, then a (directed) path inGfromxtoyof length Nis a sequence (xi)ii==N ofN+ vertices such thatx=x,xN =yand (xn–,xn)∈E(G) for i= , . . . ,N. A graphGis connected if there is a directed path between any two vertices.Gis weakly connected ifGis connected. IfGis such thatE(G) is symmetric andxis a vertex in G, then the subgraphGxconsisting of all edges and vertices which are contained in some path beginning atxis called the component ofGcontainingx. In this caseV(Gx) = [x]G, where [x]Gis the equivalence class of the following relationRdefined onV(G) by the rule:
yRzif there is a (directed) path inGfromytoz.
ClearlyGxis connected.
3 Monotone nonexpansive mappings
Throughout we assume that (X, · ) is a Banach space andτ is a Hausdorff topological vector space topology onXwhich is weaker than the norm topology. LetCbe a nonempty, convex and bounded subset ofXnot reduced to one point. LetGbe a directed graph such thatV(G) =CandE(G)⊇. Assume thatG-intervals are convex. Recall that aG-interval is any of the subsets [a,→) ={x∈C; (a,x)∈E(G)}and (←,b] ={x∈C; (x,b)∈E(G)}for anya,b∈C.
Definition . LetCbe a nonempty subset ofX. A mappingT:C→Cis called () G-monotone if(T(x),T(y))∈E(G)whenever(x,y)∈E(G)for anyx,y∈C; () G-monotone nonexpansive ifTisG-monotone and
T(x) –T(y)≤ x–y, whenever(x,y)∈E(G)
for anyx,y∈C.
Remark . For examples of metric spaces endowed with a graph andG-monotone map-pings which are Lipschitzian with respect to the graph, we refer the reader to the examples found in [].
LetT:C→Cbe aG-monotone nonexpansive mapping. Fixλ∈(, ). Letx∈Cbe
such that (x,T(x))∈E(G). Definex=λx+ ( –λ)T(x). Since the set [x,T(x)] =
[x,→)∩(←,T(x)] is convex, thenx∈[x,T(x)],i.e., (x,x) and (x,T(x)) are inE(G).
SinceTisG-monotone nonexpansive, we get (T(x),T(x))∈E(G) and T(x) –T(x)≤ x–x.
By induction we construct a sequence{xn}inCsuch that the following hold for anyn≥: (i) xn+=λxn+ ( –λ)T(xn),
(ii) (xn,xn+),(xn,T(xn))and(T(xn),T(xn+))are inE(G),
(iii) T(xn+) –T(xn) ≤ xn+–xn.
Such a sequence is known as Krasnoselskii sequence [] (see also [–]). The following result is found in [, ].
Proposition . Under the above assumptions,we have
( +nλ)T(xi) –xi≤T(xi+n) –xi+ ( –λ)–nT(xi) –xi–T(xi+n) –xi+n
for any i,n∈N.This inequality implies
lim
n→+∞xn–T(xn)= ,
i.e.,{xn}is an approximate fixed point sequence of T.
The first part of this proposition is easy to prove via an induction argument on the indexi. As for the second part, note that{xn–T(xn)}is decreasing. Indeed we have xn+–xn= ( –λ)(T(xn) –xn) for anyn≥. Therefore{xn–T(xn)}is decreasing if and only if{xn+–xn}is decreasing, which holds since
xn+–xn+ ≤λxn+–xn+ ( –λ)T(xn+) –T(xn)≤ xn+–xn
for anyn≥. So if we assume thatlimn→+∞xn–T(xn)=R> , then we leti→+∞in the main inequality to obtain
( +nλ)R≤δ(C)
for anyn∈N, whereδ(C) =diam(C). Obviously this is a contradiction since bothλandR are not equal to .
Remark . We may letλchange withn∈N. In this case the sequence{xn}is defined by
Under suitable assumptions on the sequence{λn}, we will have the same conclusions as Proposition ., see [] for more details.
Before we state the main result of this paper, let us recall the definition ofτ-Opial con-dition.
Definition . Xis said to satisfy theτ-Opial condition if whenever any sequence{yn}in Xτ-converges toy, we have
lim sup n→+∞
yn–y<lim sup n→+∞
yn–z
for anyz∈Xsuch thatz=y.
Definition . The triple (C, · ,G) has property (P) if and only if for any sequence
{xn}n∈N in C such that (xn,xn+) ∈E(G) for any n ≥, and if a subsequence {xkn} τ-converges tox, then (xkn,x)∈E(G) for alln.
Theorem . Let X be a Banach space which satisfies theτ-Opial condition.Let C be a bounded convexτ-compact nonempty subset of X not reduced to one point.Assume that (C,·,G)has property(P)and the G-intervals are convex.Let T:C→C be a G-monotone nonexpansive mapping.Assume that there exists x∈C such that(x,T(x))∈E(G).Then
T has a fixed point.
Proof Consider the Krasnoselskii sequence{xn}, from Proposition ., which starts atx.
SinceCisτ-compact, then{xn}will have a subsequence{xkn}whichτ-converges to some
pointω∈C. By properties (ii) and (P), we get (xkn,ω)∈E(G) for anyn∈N. Consider the
type function
r(x) =lim sup n→+∞
xkn–x, x∈C.
Then Proposition . implies
lim
n→+∞xkn–x–T(xkn) –x≤ lim
n→+∞xkn–T(xkn)= .
Hence
lim sup n→+∞
T(xkn) –x=lim sup
n→+∞ xkn–x,
i.e.,r(x) =lim supn→+∞T(xkn) –xfor anyx∈C. In particular we have
rT(ω)=lim sup n→+∞
T(xkn) –T(ω)≤lim sup
n→+∞
xkn–ω=r(ω).
In fact we have r(T(x))≤r(x) for anyx∈C which is an upper-bound of{xkn}. Finally
ifXsatisfies theτ-Opial condition, then we must haveT(ω) =ω,i.e.,ωis a fixed point
Remark . The existence ofx∈Csuch that (x,T(x))∈E(G) was crucial. Indeed,
the Krasnoselskii sequence {xn} will satisfy (xn,xn+)∈E(G) for every n. However, if
(T(x),x)∈E(G) then we will have (xn+,xn)∈E(G) for everyn. In this case, we need to revise property (P) into property (P∗) defined as follows:
The triple (C, · ,G) has property (P∗) if and only if for any sequence{xn}n∈NinC
such that (xn+,xn)∈E(G) for anyn≥, and if a subsequence{xkn}τ-converges tox,
then (x,xkn)∈E(G) for alln.
The following results are direct consequences of Theorem ..
Corollary . Let C be a bounded closed convex nonempty subset of lp, <p< +∞.Letτ be the weak topology.Let G be the digraph defined on lpby({αn},{βn})∈E(G)iffαn≤βn for any n≥.Then any G-monotone nonexpansive mapping T:C→C has a fixed point provided there exists a point x∈C such that(x,T(x))∈E(G)or(T(x),x)∈E(G).
Remark . The case ofp= is not interesting for the weak-topology sincelis a Schur
Banach space. But if we consider the weak∗-topologyσ(l,c) onlor the pointwise
con-vergence topology, thenlsatisfies the weak∗-Opial condition. In this case we have a
sim-ilar conclusion of Corollary . forl.
Corollary . Let C be a bounded closed convex nonempty subset of Lp, ≤p< +∞.Let
τ be the almost everywhere convergence topology.Let G be the digraph defined on Lp by (f,g)∈E(G)if and only if f(t)≤g(t)almost everywhere.Assume that C is almost every-where compact.Then any G-monotone nonexpansive mapping T:C→C has a fixed point provided there exists a point f∈C such that(f,T(f))∈E(G)or(T(f),f)∈E(G).
Competing interests
The author declares that he has no competing interests.
Acknowledgements
The author acknowledges King Fahd University of Petroleum and Minerals for supporting this research. The author thanks the referees for their valuable suggestions including the reference [5].
Received: 5 January 2015 Accepted: 19 March 2015 References
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